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Number of integer factorizations of n into nonsquarefree factors > 1.
2

%I #10 Oct 08 2024 18:39:44

%S 1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,2,0,0,

%T 0,2,0,0,0,1,0,0,0,1,1,0,0,2,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,4,0,0,0,1,

%U 0,0,0,3,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0

%N Number of integer factorizations of n into nonsquarefree factors > 1.

%e The a(n) factorizations for n = 16, 64, 72, 144, 192, 256, 288:

%e (16) (64) (72) (144) (192) (256) (288)

%e (4*4) (8*8) (8*9) (4*36) (4*48) (4*64) (4*72)

%e (4*16) (4*18) (8*18) (8*24) (8*32) (8*36)

%e (4*4*4) (9*16) (12*16) (16*16) (9*32)

%e (12*12) (4*4*12) (4*8*8) (12*24)

%e (4*4*9) (4*4*16) (16*18)

%e (4*4*4*4) (4*8*9)

%e (4*4*18)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],NoneTrue[SquareFreeQ]]],{n,100}]

%Y For prime-powers we have A000688.

%Y Positions of zeros are A005117 (squarefree numbers), complement A013929.

%Y For squarefree instead of nonsquarefree we have A050320, strict A050326.

%Y For nonprime numbers we have A050370.

%Y The version for partitions is A114374.

%Y For perfect-powers we have A294068.

%Y For non-perfect-powers we have A303707.

%Y For non-prime-powers we have A322452.

%Y The strict case is A376679.

%Y Nonsquarefree numbers:

%Y - A078147 (first differences)

%Y - A376593 (second differences)

%Y - A376594 (inflections and undulations)

%Y - A376595 (nonzero curvature)

%Y A000040 lists the prime numbers, differences A001223.

%Y A001055 counts integer factorizations, strict A045778.

%Y A005117 lists squarefree numbers, differences A076259.

%Y A317829 counts factorizations of superprimorials, strict A337069.

%Y Cf. A008480, A053797, A053806, A061398, A089259, A120992, A124010, A182853, A373198, A375707, A376312.

%K nonn

%O 1,16

%A _Gus Wiseman_, Oct 07 2024